Mathematics for the interested outsider

The Inverse Function Theorem

Recall the inverse function theorem from multivariable calculus: if is a map defined on an open region , and if the Jacobian of has maximal rank at a point then there is some neighborhood of so that the restriction is a diffeomorphism. This is slightly different than how we stated it before, but it’s a pretty straightforward translation.

Anyway, this generalizes immediately to more general manifolds. We know that the proper generalization of the Jacobian is the derivative of a smooth map , where is an open region of an -manifold and is another -manifold. If the derivative has maximal rank at , then there is some neighborhood of for which is a diffeomorphism.

Well, this is actually pretty simple to prove. Just take coordinates at and at . We can restrict the domain of to assume that is entirely contained in the coordinate patch. Then we can set up the function .

Since has maximal rank, so does the matrix of with respect to the bases of coordinate vectors and , which is exactly the Jacobian of . Thus the original inverse function theorem applies to show that there is some on which is a diffeomorphism. Since the coordinate maps and are diffeomorphisms we can write for some , and conclude that is a diffeomorphism, as asserted.

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This is mainly an expository blath, with occasional high-level excursions, humorous observations, rants, and musings. The main-line exposition should be accessible to the “Generally Interested Lay Audience”, as long as you trace the links back towards the basics. Check the sidebar for specific topics (under “Categories”).

I’m in the process of tweaking some aspects of the site to make it easier to refer back to older topics, so try to make the best of it for now.